MAINTAINING UNIFORM DENSITY AND MINIMIZING THE CHANCE OF ERROR IN A LARGE SCALE WSN

International Journal of Network Security & Its Applications (IJNSA), Vol.3, No.3, May 2011
DOI : 10.5121/ijnsa.2011.3307 98
MAINTAINING UNIFORM DENSITY AND
MINIMIZING THE CHANCE OF ERROR IN A LARGE
SCALE WSN
Asis Kumar Tripathy1
,Shradhananda Beura2
,Muralidhar Behera3
and Gunanidhi
Pradhan4
1,2,3Department of Computer Science and Engineering, NM Institute of Engineering and Technology, Bhubaneswar,Odisha,India
{1
asistripathy,2
beura.shradhananda,3
behera.muralidhar}@gmail.com
4Govt. college of Engineering, Kalahandi, Odisha,India
4
gunanidhipradhan@gmail.com
ABSTRACT
In a real application area, the WSN is not a homogeneous network where the nodes are maintained in
respective coordinate position relatively same to each other. But rather homogeneous it should be
heterogeneous, where the relative positional difference for different nodes are different. In this paper a
better scheme is being proposed which will take care of the life time and density of a WSN. Sun et. al.
proposed uniform density in WSN by assuming the network as a homogeneous network ,but in this paper
without taking a homogeneous network the same problem is being solved by using the Gaussian
probability density function. And also the chance of error in receiving the message from the WSN to the
base station is minimized by using priori probability algorithm.
KEYWORDS
WSN, Network density, life time, priori probability
1. INTRODUCTION
Sensor networks are composed of small electronic devices, the sensors that monitor areas,
objects, animals, persons or sense temperature, humidity, the presence of acoustic or seismic
waves etc. in a given area of interest. WSN can be used for remote monitoring and object
tracking in different environments and for a wide range of applications. Thus recent advances
in computing hardware and software are responsible for the emergence of sensor networks
capable of observing the environment, processing the data and making decisions based on the
environment, detect and locate specific events and track targets over a specific regions. Thus a
sensor network is defined as being composed of a large number of nodes which are deployed
densely in close proximity to the phenomenon to be monitored. Each of these nodes collect
data and its purpose is to route this information back to a sink. Most previous papers
concentrate in homogeneous WSN in many application areas. But practically in many areas
homogeneous WSN deployment is not possible. That is the network is to be deployed can be
heterogeneous WSN. So, in heterogeneous network the network density over a deployed area
may not be uniform. Because in a specified area the position of deployed sensor nodes cannot
be maintained in the homogeneous coordinate separation. That is the area covered by the each
sensor node’s signal is not uniform for the entire deployed sensor node due to the positional

99
difference is different for every adjacent sensor nodes out of all nodes in WSN. This is one
constraint for heterogeneous WSN in real application field.
Another constraint in WSN is the definition of transmitted message from the nodes t the sink
node, last to the processing device may not be error free due to the attenuation caused by
environment in any intentional man made. That is if noise is added to the message during
transmission then the message which has been sent from the WSN to the receiver may not be
identified as the correct message successfully, due to the addition of noise during transaction
period which leads receiving and processing of an erroneous message and the actual sensing
event cannot be predicted though it has been censored by the WSN. That’s why an algorithm
should be developed which can be supported by a specific device and can receive and identify
the same message which has been transmitted. This can be done by likelihood probability
method called port priori algorithm in which the message received is error free.
2. RELATED WORK
Sun et. Al’s proposed scheme is based on the homogeneous WSN, where different nodes are
deployed with in a predetermined area and the sink node is placed at the coordinate (0,0) of that
specified area of the WSN[16]. This paper concentrates on different energy consumption in the
network, when the network density is uniform. i.e. the information detected will be forwarded
to the sink node so that the energy of the nodes which are closed to the sink will be burnt faster.
Once these nodes died, the life time is over the energy of the nodes that are far away from the
sink node will be watched, which is inefficient. The waste of energy will be concentrated on
this paper to leave nearly no energy when life time comes to its limit, thus makes the network
provide service oriented.
The lifetime limits of WSN with and without aggregation have been taken into consideration by
Manish Bhardwaj, Anantha P. and Chandrakasan, which gives us the feasibility to do some
further work on lifetime[2,4]. After that, a lot of work has been carried out to deal with the
lifetime of WSN, the principal goal of which is the efficient use of energy[5-8]. These
researches aim at modeling the lifetime and finding a method to optimize it. These work have
been done based on the existing definitions of lifetime[9,10]. But on some occasions these
standards are inappropriate. The diversity of source behavior and the layout of the network are
important factors for they can take effect once the network is deployed[11-14]. Some research
work has been done to prolong the lifetime of WSN via unequal clustering or scheduling the
selection of cluster heads [14,15]. But the solution should be more active, the layout should be
well deployed to create wider potential space for optimizing the lifetime before the network
takes into action. Different nodes within the network have different service requirements, which
inspires the idea of this paper, that is, network lifetime should be service-oriented. A service-
oriented assessment of lifetime is explored in this paper. It is investigated from two
perspectives, one is how to assess network lifetime from service-oriented aspect, and the other
is how to deploy more nodes to balance the energy consumption where more service is needed.
3. OVER VIEW OF SUN ET. AL’S SCHEME
In this paper there are various definitions are presented.

100
definition1: Network density is defined s, in a specified area A where, nodes are deployed,
),( yxρ is defined as the network density with the property
1),( =∫∫A
dxdyyxρ
i.e the network
is assumed as homogeneous.
definition2: Network energy intensity is given and defined as e(x,y)=e0N
∫∫D
dxdyyx ),(ρ
where, e(x,y) is defined as network energy intensity with initial energy 0e
, for the given region
D. With the definition of network density relying models are specified for a routing algorithm
ψ. The relying probability makes the conditional probability density function Rψ
(x,y,a,b,ρ(x,y)), with the property
1)),(,,,,( ≥∫∫A
dxdyyxbayxR ρψ
, where Rψ
(x,y,a,b,ρ(x,y)) denotes the probability of (x,y) that will do the rely if there is an event
generated at (a,b).The partial density function of the sourced is employed S(x,y) with the
property
∫∫ =
A
MdxdyyxS ),(
Where S(x,y) denoted the distribution of source behavior with in
a period T and M denotes the intensity of source behavior.
definition 3: Network requesting intensity C(x,y) is defined as :
C(x,y)=
∫∫A
dadbbaSyxbayxR ),()),(,,,,( ρλ ψ
Where λ is a constant number.
Theorem-1The energy wasted is quantized by the expression.
∫∫A
dxdyyxC ),(
with e(x,y)=T(C(x,y)∀ (x,y)) where T is a constant value.
Section 4: Quantization of network density is explained by the density iterative process.
Step 1. i=0; Let the network be uniformly deployed first, then the density distribution
ρ is
obtained, which is a constant function.
Step 2. After one period T of data gathering the distribution of energy consumption denoted by
network requesting intensity ( , ) i c x y is derived.
Step 3. Use ( , ) i c x y derived in step 2 to guide the new-round deployment of network density.
In section 5, Stability and uniform convergence of the network is analyzed with the equation.
=),( yxρ
dxdydadbbaSyxbayxR
dadbbaSyxbayx
A A
∫∫ ∫∫
∫∫








),()),(,,,,(
),()),(,,,,(R
A
ρ
ρ
ψ
ψ

101
Theorem 2:
It has been explained that.
=),( yxei
dxdydadbbaSyxbayxR
dadbbaSyxbayx
A A
i
i
∫∫ ∫∫
∫∫








−
−
),()),(,,,,(
),()),(,,,,(R
1
A
1
ρ
ρ
ψ
ψ
If 1<γ by taking
Rψ (x,y,a,b,
),(.),(,,,,()),(,,,,( 0 yxyxbayxRyxbayxR ii ργηρρ ψψ −+=
with
0),(.- i =∫∫ dxdyyx
A
ργη
Theorem-3
It has been explained that when
∫∫A
dadbbaSyxbayxR ),()),(,,,,( ρψ
it is not a constant
function then it will dip get uniform convergence.
So over ally, the research on this service oriented network density WSN paper, the network is
considered as homogeneous and the density iterative process is prepared to quantize the
network density which can enlarge the potential space for optimizing the lifetimes.
Practically when WSN is to be deployed it, can’t be homogenous. So in this paper practical
approach has been explained where a non-homogeneous i.e heterogeneous WSN can be
feasible with the possible error elimination, when message is transmitted. Also it has been
proved simulated that, for uniform network the network density ),( yxρ is with a property
1),( ≠∫∫A
dxdyyxρ
4. PROBLEM DEFINITION
Section-1
In this scheme, in definition(1), network density is considered and it is denoted as f(x,y). Here
as the WSN is considered as homogeneous, therefore the network has taken as uniform network
where the network density distribution taken as
),(),( 0 yxyx ρρ =
which is a constant function
with a property
1),( =∫∫A
dxdyyxρ

102
For a homogeneous WSN where nodes are deployed uniformly, then the possible schematic
diagram of the network will be as follows,
When the network is uniform i.e the position difference every two consecutive nodes out of all
nodes should be same. This is also is represented in the above diagram.
We can assume, the network is uniform if, Positional coordinates(Nj) - Positional
coordinates(Ni) = constant ∀ N∈WSN deployed
Nj(x,y)-Ni(x,y)= C ∀ N∈WSN deployed where, C is a constant. Suppose the practical area
covered by the signals of a specific deployed node inside homogeneous network area is,
ap(Nk(x,y) is same for all n nodes ap
⇒
∑=
n
k
kN
1
p )(a
=A, where A is the whole area of the WSN.
Then for a uniform WSN,
1),)(( =∫∫A
kp dxdyyxNa
But practically a network can not be uniform because when a network is to be established by
deploying the nodes over a specified area, then the nodes can not be deployed uniformly over
that specified area.
i.e Nj(x,y)-Ni(x,y) ≠ C ∀ N∈WSN deployed
Z
Y
x
(0,0) Sink node

103
Practically nodes are deployed non-uniform over a specified area. The likely hood schematic
diagram is as follows,
therefore ⇒
∑=
≠
n
k
k AN
1
p )(a
That means, there exist some partial areas which are not covered by the sensor node; signal.
Those are out of coverage of the signal.
Section-2
What happened for the area out of coverage of the signal?
First considering the uniform network, with constant network density ρ(x,y), there is a routing
algorithm ψ, the relying probability matches the conditional probability density function Rψ
(x,y,a,b,ρ(x,y)), with the property
1)),(,,,,( ≥∫∫A
dxdyyxbayxR ρψ
, where
Rψ (x,y,a,b,ρ(x,y)) denotes the probability of (x,y) that will do the relying if there is an event
generated at (a,b).
Suppose that event generated at (a,b) is now with in the out of network coverage area of WSN.
Then the property of routing algorithm, because the generated event may not be predicted.
i.e.
1)),(,,,,( <∫∫A
dxdyyxbayxR ρψ
Z Y
x
(0,0) Sink node

104
in theorem 2 it is taken as, Rψ (x,y,a,b,ρi(x,y))= Rψ (x,y,a,b,ρ0(x,y)) +η - γ . ρi(x,y)
where
0),(.- i =∫∫ dxdyyx
A
ργη
where ρi(x,y)
=
dxdydadbbaSyxbayxR
dadbbaSyxbayx
A A
i
i
∫∫ ∫∫
∫∫








−
−
),()),(,,,,(
),()),(,,,,(R
1
A
1
ρ
ρ
ψ
ψ
Where S(a,b) the spatial probability density function of a source.
ρi-1(x,y) the network density distribution of the preceding node Ni and this ρi(x,y)
converges if 1<γ .
But for a non-uniform network, it may not be converged for 1<γ , because 1<γ
1)),(,,,,( <∫∫A
dxdyyxbayxR ρψ
For uniform network the network density is ρ(x,y) with a property
∫∫ =
A
dxdyyx 1),(ρ
But for
practical non-uniform network, using Gussian probability density we can perform a number of
experiments to verify that
∫∫ ≠
A
dxdyyx 1),(ρ
The Gaussian probability density function is f(x)=
2
2
1
πδ e
2
2
2
)(
δ
mx−−
Where, m and
2
δ are the average value and the variance associated with Gaussian probability
density function f(x).
Thus X =
dx
xe
mx
∫
∞
∞−
−−
2
2
)(
2
2
2
πδ
δ
= m

105
Now the cumulative distribution corresponding to the Gaussian probability density for m=0, is
p(X≤x)=F(x)=
dx
xe
x
x
∫∞−
−
2
2
2
2
2
πδ
δ
Now, the error function of u is given by,
erf u=
due
u
u
∫
−
0
22
π
the complementary error function is denoted as erfc u and given by,
erfc u=1-erf u
=
due u
∫
∞
−
0
22
π
Now the cumulative distribution function F(x) is expressed in terms of the error function and
the complementary function, for x≥0,
F(x) =
dx
xe
x
x
∫∞−
−
2
2
2
2
2
πδ
δ
=
dx
xe
x
∫
∞
∞−
−
2
2
2
2
2
πδ
δ
-
dx
xe
x
x
∫
∞
−
2
2
2
2
2
πδ
δ
For x≤0, for positive u,
F(x)=F(-|x|) =
dx
xe
x
x
∫
−
∞−
−
||
2
2
2
2
2
πδ
δ
=
due
x
u
∫
−
∞−
−
π
π
2
||
21
Let u−=ε
F(x) =










∫
∞
−
ε
π
δ
ε
de
x
2
||
22
2
1
=






δ2
||
2
1 x
erfc

106
Now, for a constant number k, the probability on the standard deviation δ ,
P±kδ = P(m-kδ ≤X≤m+kδ ) = 2
k
erf
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
k
Probabilityofnetworkdensity
Section-3
Error calculation:
As the signal traverses the transmission medium noise will be added to the signal. Hence as
finally received and pre received the signal will be corrupted or contaminated by noise from a
number of sources.
Let us consider one case:
In successive intervals we want to transmit one of the two possible messages. The two possible
messages might be represented at the transmitting end by two distinct wave form, each limited
in time period to the interval allocated to a bit. At the receiving end we right desire a system
where by the message m0 when received, generates some voltage r0, for m1 received generates
voltage r1 in the absence of noise. Because of noise, error occurred i.e for message m0
received indication might be r1 and for m1 r0
We shall assume, for generality, that the probability of an error is dependant on which msg was
send and transition probabilities are introduced. i.e
P(r0 | m0) = probability that r0 is received for m0 sent.

107
Priori probabilities:
For the general case, messages m1 and m0 do not occur with equal frequency and probabilities
p(m1) and p(m0) are introduced. Probabilities that m1 and m0 intended in an arbitrary message
interval. These probabilities p(m1) and p(m0) are called priori probabilities.
Now from an observed response r1 and r0, it can not be determined with certainty which of the
message was sent. Now by posteriori probability (message intended with maximum probability
that our opinion is correct).
If r0 is received
Choose m0 if P(m0 | r0) > P(m1 | r0)
If r1 is received
Similarly,
If r0 is received, m0 should be chosen if
P(m0 | r0)P(r0) > P(m1 | r0) P(r0)
Correspondingly If r1 is received, m1 should be chosen if
P(r1 | m1)P(m1) > P(r1 | m0) P(m0)
This is optimum – receiver algorithm.
In general, there can be k messages m1,m2,…….mk and j received rensposes r1,r2,………. rj ,
by optimum-receiver algorithm, if rj is received, choose mk if
P(mk | rj) > P(mi | rj) ∀ i≠k.
Optimum receiver:
A receiver which operates in accordance with this algorithm to maximize a posteriori
probability of a correct decision and is called an optimum receiver.

108
6.CONCLUSION
In this paper , it is more concentrated on the density of the network in a heterogeneous WSN. A
method to assess lifetime from the perspective of service is first presented, in which whether
the network can support service is concentrated on and also the chance of error at the receiving
site is strictly minimized. This is also being analyzed that how the sink node will be used for
maximum time then also there will not be any problem because of the lifetime of the sensors.
Deployment can be guided to enlarge the potential space for optimizing the lifetime of the
network. The Proposed method is also practically being analyzed the feasibility of the network.
REFERENCES
[1] Akyildiz I.F, Su W, etc. Wireless sensor network: A survey [J]. Computer Networks. 2002,
38(4): 393-422.
[2] Bhardwaj M, Chandrakasan A, Garnett T. Upper bounds on the lifetime of sensor networks[C].
In: IEEE Int’l Conf. on Communications. Helsinki: IEEE Computer Society, 2001, 785-790.
[3] Duarte-Melo E. J, Liu M, Misra A. A Modeling Framework for Computing Lifetime and
Information Capacity in Wireless Sensor Networks[C]. In: Modeling and Optimization in
Mobile, Ad Hoc and Wireless Networks, Cambridge, UK, March 2004.
[4] Bhardwaj M, Chandrakasan A.P. Bounding the Lifetime of Sensor Networks Via Optimal Role
Assignments [C]. In: INFOCOM. New York, USA: IEEE, 2002. 1587-1596.
[5] Kalpakis K. Efficient Algorithms for Maximum Lifetime Data Gathering and Aggregation in
Wireless Sensor Networks [J]. Computer Networks: The International Journal of Computer and
Telecommunications Networking. August 2003, 42(6): 697-716.
[6] Enrique J. Duarte-Melo, Mingyan Liu. Analysis of Energy Consumption and Lifetime of
Heterogeneous Wireless Sensor Networks[C]. In: Proc. of the GLOBECOM 2002. New York:
IEEE Press, 2002. 21-25.
[7] Rai V, Mahapatra R. N. Lifetime Modeling of a Sensor Network[C]. In: Design , Automation
and Test in Europe, Munich , Germany ,March 2005. 202-203.
[8] Yuan Xue, YI CUI, etc. Maximizing Lifetime for Data Aggregation in Wireless Sensor
Networks[J]. Mobile Networks and Applications. 2006, 10(6): 853-864.
[9] A. Ephremides. Energy Concerns in Wireless Networks [J]. IEEE Wireless Communications,
vol. 9, pp. 48-59, Aug. 2002.
[10] Jianping Pany Y. Thomas Houz Lin Caiy etc. Topology Control for Wireless Sensor Networks
[C]. In: nternational Conference on Mobile Computing and Networking. 2003, 286-299.
[11] Egorova-Forster, Murphy. Exploring Non Uniform Quality of Service for Extending WSN
Lifetime[C]. In: Fifth IEEE International Conference on Pervasive Computing and
Communications Workshops (PerComW'07) 2007. 285-289.
[12] Lu Ke zhong, Huang Liu sheng, etc. Deploying Sensor Nodes in Wireless Sensor Networks[J].
Journal of Chinese Computer Systems. 2006, 27(11):2003-2006.
[13] Guihai Chen. Chengfa Li. An Uneven Cluster-Based Routing Protocol for Wireless Sensor
Networks [J]. Chinese Journal of Computers. 2007, 30(1):27-36

109
[14] Soro S, Heinzelman WB. Prolonging the lifetime of wireless sensor networks via unequal
clustering[C]. In: Proc. of the 19th IEEE Int'l on Parallel and Distributed Processing
Symposium. San Francisco: IEEE Computer Society Press, 2005. 236-240.
[15] Biswanath Dey Sukumar Nandi. Distributed Location and Lifetime Biased Clustering for Large
Scale Wireless Sensor Network[C]. In : Distributed Computing and Networking, 8th
International Conference, ICDCN 2006. 534-54
[16] Dayang Sun,Yanheng liu,Aimin wang, Bin Ge Research on Service-oriented Lifetime and
Network Density in WSN In:IEEE 9th International Conference for Young Computer Scientists
2008, 439-444
[17] Communication System, Taub & Sheiling.
Authors
Asis Kumar Tripathy Received Diploma in IT in 2002, B.E. IT in 2006 and MTech
in Comp. Sc. & Engg. In 2010 from Inter-National Institute of Information
Technology, Bhubaneswar. He has Joined NM Institute of Engg. & Technology as a
lecturer from 01-07-2010.His research areas are Network Security and Wireless
Sensor Network.
Shradhananda Beura received Diplma in IT, BTech. From CET, Bhubaneswar In
2004 & 2008 respectively. He is persuing MTech. In CSE. His research area is
Wireless Sensor Network.
Muralidhar Behera received MCA in 2003, Persuing MTech. In CSE. His research area is Wireless
Sensor Network.
Dr. Gunanidhi Pradhan Received Ph.D in Information & Communication
Technology.
Awarded Best Teacher Award by ISTE,Currently Principal, Govt. College of
Engineering, Kalahandi,Bhawanipatna,Odisha

MAINTAINING UNIFORM DENSITY AND MINIMIZING THE CHANCE OF ERROR IN A LARGE SCALE WSN

Recommended

Recommended

More Related Content

What's hot

What's hot (13)

Similar to MAINTAINING UNIFORM DENSITY AND MINIMIZING THE CHANCE OF ERROR IN A LARGE SCALE WSN

Similar to MAINTAINING UNIFORM DENSITY AND MINIMIZING THE CHANCE OF ERROR IN A LARGE SCALE WSN (20)

Recently uploaded

Recently uploaded (20)

MAINTAINING UNIFORM DENSITY AND MINIMIZING THE CHANCE OF ERROR IN A LARGE SCALE WSN